Question

Find a recursive formula for the sequence:

Answer 1

Explanation:

`a _(n) = - 2a _(n - 1) + 1`

Answer 2

`a_n=-2a_(n-1)+1`

Explanation:

Successive terms do not have a common difference or ratio, so the sequence is not arithmetic or geometric. We can get a clue as to its nature a couple of places:

1. from the alternating signs, says there is a negative multiplier involved
2. from the answer choices

It is helpful to look at the differences of terms:

-5-3 = -8

11 -(-5) = 16

-21-11 = -32

These have a common ratio of -2, so match an exponential form with a ratio of -2 (the last answer choice). Trying that choice with n=1, 2, 3, 4 gives ...

a[1] = 3

a[2] = -2(3) +1 = -5

a[3] = -2(-5) +1 = 11

a[4] = -2(11) +1 = -21

These match the given sequence, so the appropriate recursive formula is ...

`\boxed{a_n=-2a_(n-1)+1}`